An Interpolation Problem by Completely Positive Maps and its. Quantum Cloning

Transcription

1 An Interpolation Problem by Completely Positive Maps and its Application in Quantum Cloning College of Mathematics and Information Science, Shaanxi Normal University, Xi an, China, July 20, 2011

2 Introduction The theory of quantum information is a result of the effort to generalize classical information theory to the quantum world. Quantum information theory aims to answer the following question: What happens if information is stored in a state of a quantum system? To characterize the way the information is processed consistent with the laws of quantum mechanics, in mathematics, a quantum operation is a trace preserving completely positive map from the set of density operators into itself. The completely positive maps play an important role in the description of quantum channels and time evolutions and have triggered an increasing interest.

3 Outline 1 Notations 2 An interpolation question 3 Some applications

4 Question Denote by M M,N the set of all M N complex matrices, for brevity, write M N when M = N and let H N be the set of Hermitian matrices in M N. From C. K. Li and Y. T. Poon, Interpolation problems by completely positive maps, arxiv: v1. Question Given {A i } k i=1 M N and {B i } k i=1 M M, determine the necessary and sufficient condition for the existence of a completely positive map φ : M N M M such that φ(a i ) = B i for every 1 i k, and possibly with the additional properties that φ is unital or/and φ is trace preserving.

5 Question Theorem. Let {A i } k i=1 H N and {B i } k i=1 H M be two commuting families. Then there exist unitary matrices U M N and V M M such that U A i U and V B i V are diagonals a i b i a i b i and, a in b im respectively, for i = 1, 2,..., k. Then the following conditions are equivalent. (a) There is a completely positive map φ : M N M M such that φ(a i ) = B i for every 1 i k.

6 (b) There is an N M nonnegative matrix M such that B = AM, where b 11 b b 1M a 11 a a 1N b 21 b b 2M a 21 a a 2N B =, A = b k1 b k2... b km a k1 a k2... a kn Furthermore, (1) φ in (a) is unital if and only if M in (b) can be chosen to be column stochastic; (2) φ in (a) is trace preserving if and only if M in (b) can be chosen to be row stochastic; (3) φ in (a) is unital and trace preserving if and only if M in (b) can be chosen to be doubly stochastic.

7 Notations Consider an N M rectangular matrix A = [a ij ], we define the reshaping of A as res(a) = (a 11,..., a 1M, a 21,..., a 2M,..., a N1,..., a NM ) t, (2.1) where X t denotes the transpose of a matrix X. Clearly, the length of res(a) is NM. Conversely, any vector of length NM may be reshaped into an N M rectangular matrix. For example, ( a1 a 2 ) (a 1, a 2, a 3, a 4 ) a 3 a 4.

8 Notations For arbitrary MN ST matrix C, usually we use both indices to denote C = [a ij ], but more conveniently, we will use the blockform of C as C = [C m,n ] M M,N (M S,T ) and then write as C = [C m,n ] = [cm,µ], (2.2) n,ν where cm,µ n,ν denotes the (µ, ν)-entry of (m, n)-block matrix C m,n.

9 Notations For example, ( C11 C 12 C = C 21 C 22 ) = c c c c c c c c c c c c = [cm,µ n,ν ]. c c c c Especially, when C = A B, A = [a mn ] M M,N, B = [b µν ] M S,T, we have cm,µ n,ν = a mnb µν.

10 Notations For arbitrary MN ST matrix C defined by Eq.(2.2), we define a block-matrix where cm,n µ,ν For instance, if C = C κ = [Cm,µ] κ = [cm,n], (2.3) µ,ν denotes the (n, ν)-entry of (m, µ)-block matrix Cm,µ. κ ( C11 C 12 C 21 C 22 ) = c c c c c c c c c c c c c c c c,

11 Notations then C κ = c c c c c c c c c c c c c c c c.

12 Notations C = z z z 1,3 z z z 2,3 z z z 1,3 z z z 2,3 z1,3 z1,3 z1,3 1,3 z1,3 z1,3 z1,3 2,3 z z z 1,3 z z z 2,3 z z z 1,3 z z z 2,3 z2,3 z2,3 z2,3 1,3 z2,3 z2,3 z2,3 2,3, C κ = z z z 1,3 z z z 1,3 z1,3 z1,3 z1,3 1,3 z z z 2,3 z z z 2,3 z1,3 z1,3 z1,3 2,3 z z z 1,3 z z z 1,3 z2,3 z2,3 z2,3 1,3 z z z 2,3 z z z 2,3 z2,3 z2,3 z2,3 2,3.

13 Notations Let C be as in Eq.(2.2). Then (C κ ) κ = C. Theorem [Choi, 1975] Let φ : M N M M be a linear map. Then φ is completely positive if and only if φ is of the form φ(ρ) = k i=1 V iρv i for all ρ in M N where V i is an M N matrix for each 1 i k. Remark Note that {V i } k i=1 is not uniquely determined by φ. If there exists another family {W i } l i=1 such that φ(ρ) = l i W iρw i for all ρ in M N, then there must be a unitary U = [u ij ] such that W i = k j=1 u ijv j for every 1 i l.

14 Notations For a completely positive map φ : M N represented as M M, let it be φ(ρ) = k i=1 V i ρv i, ρ M N, (2.4) Let the M N matrix V i satisfy Eq.(2.4) be written as V i = [a i mn] for i = 1, 2,..., k. Define a Kraus matrix M of φ as [ k M = a i mn 2]. (2.5) i=1

15 Notations Remark In fact, M = k i=1 [ai mn] [a i mn], where denotes the Hadamard product of V i and V i, a i mn is the conjugate of a i mn. Even though Kraus operators of φ are not unique, by Remark 1 and Eq. (2.5), the Kraus matrix M is unique and written M φ, clearly, it is a nonnegative(its entries 0) matrix.

16 Notations Now we reshape V i into a length MN column vector res(v i ) ( ) following Eq.(2.1). Define V := res(v 1 ), res(v 2 ),..., res(v k ), and put D = VV, namely, a 1 11 a a k a 1 1N a 2 1N... a k 1N V = , a 1 M1 a 2 M1... a k M a 1 MN a2 MN... ak MN which is an MN k matrix, and

17 Notations D = k i=1 ai 11 ai 11 k i=1 ai 11 ai k i=1 ai 11 ai MN k i=1 ai 1N ai 11 k i=1 ai 1N ai k i=1 ai 1N ai MN k i=1 ai M1 ai 11 k i=1 ai M1 ai k i=1 ai M1 ai MN k i=1 ai MN ai 11 k i=1 ai MN ai k i=1 ai MN ai MN, which is an MN MN matrix. (2.6)

18 Notations Let zm,µ n,ν D = = k i=1 ai mna i µν. Then D can be written as follows z... z... z1,m... z1,m 1,N 1,N z... z1,m N,N N, z N,1 zm,1... z1,m N,N... zm,1... zm,m 1,N zm,1 N,1... zm,1 N,N... zm,m N,1... zm,m 1,N... zm,m N,N. (2.7)

19 Notations Also, D can be represented by following formalism D = k i=1 ( )(. res(v i ) res(v i )) (2.8) The following gives a matrix representation of Eq.(2.4). Proposition If φ satisfies Eq.(2.4), then res(φ(ρ)) = D κ φ res(ρ) for each ρ M N.

20 An interpolation question Theorem (1) If φ : M N M M is a completely positive map, then D φ M MN is positive semi-definite. (2) If D M MN is positive semi-definite, then there exists a completely positive map φ : M N M M such that D φ = D. (3) Suppose that φ is a completely positive map, then (i) φ is unital if and only if tr 2 D φ := N µ=1 z m,µ = I n,µ M; (ii) φ is trace preserving if and only if tr 1 D φ := M m=1 z m,ν = I m,µ N.

21 An interpolation question Remark The matrix M in (b) of Theorem in the beginning is just transpose M t φ of the Kraus matrix of the completely positive map φ in (a). The following theorem is an our main result considering the existence of completely positive maps φ sending a general family of matrices in M N to another family in M M.

22 An interpolation question Theorem Let {A i } k i=1 M N and {B i } k i=1 M M. Then the following conditions are equivalent. (a) There exists a completely positive map φ : M N M M such that φ(a i ) = B i for every 1 i k. (b) There exists an MN MN positive semi-definite matrix E such that (res(b 1 ), res(b 2 ),..., res(b k )) = E κ (res(a 1 ), res(a 2 ),..., res(a k )). (3.1)

23 An interpolation question Moreover, if one of (a) and (b) holds, then (1) φ in (a) is unital if and only if E in (b) satisfies tr 2 E = I M ; (2) φ in (a) is trace preserving if and only if E in (b) satisfies tr 1 E = I N. (3) φ in (a) is unital trace preserving if and only if E in (b) satisfies tr 1 E = I N and tr 2 E = I M.

24 An interpolation question Corollary Let {A i } k i=1 M N and {B i } k i=1 M M be two families consisting of diagonal matrices, for every 1 i k, denote a i b i a i b i A i = and B i = a in b im Then the following conditions are equivalent.

25 An interpolation question (a) There is a completely positive map φ : M N M M such that φ(a i ) = B i for every 1 i k. (b) There is an N M nonnegative matrix M such that B = AM, where b 11 b b 1M a 11 a a 1N b 21 b b 2M a 21 a a 2N B =, A = b k1 b k2... b km a k1 a k2... a kn (c) There exists a positive semi-definite matrix E with size MN satisfying Eq.(3.1).

26 Applications Denote by D N the set of all density matrices in M N. Definition Let {ρ i } k i=1 D N. If there exists a quantum operation φ : M N M N M N M N and a standard state Σ D N such that φ(ρ i Σ) = ρ i ρ i, i = 1, 2,..., k, then we call {ρ 1, ρ 2,..., ρ k } is clonable.

27 Applications It was essentially proved in [H. Barnum, C. M. Caves, et al., Noncommuting Mixed States Cannot Be Broadcast] that {ρ 1, ρ 2 } is clonable if and only if ρ 1 and ρ 2 are identical or orthogonal (i.e. ρ 1 ρ 2 = 0). The following theorem generalizes this conclusion, which will be proved with a very simple way that is different from the old approach. Let we see the following properties of fidelity F (ρ 1, ρ 2 ) of two arbitrary density matrices ρ 1, ρ 2 are needed.

28 Applications Lemma For all density matrices ρ i D N (i = 1, 2, 3, 4), the following are valid: (1) 0 F (ρ 1, ρ 2 ) 1; (2) F (ρ 1, ρ 2 ) = 0 if and only if ρ 1 ρ 2 = 0; (3) F (ρ 1, ρ 2 ) = 1 if and only if ρ 1 = ρ 2 ; (4) F (ρ 1 ρ 2, ρ 3 ρ 4 ) = F (ρ 1, ρ 3 )F (ρ 2, ρ 4 ).

29 Applications Theorem Let {ρ i } k i=1 D N and ρ i ρ j if i j. Then {ρ i } k i=1 and only if ρ i ρ j = 0 for 1 i j k. is clonable if

30 Applications Proof. Necessity. Suppose that {ρ i } k i=1 is clonable, then we see that there exists a quantum operation φ : M N M N M N M N and a standard state Σ such that φ(ρ i Σ) = ρ i ρ i, i = 1, 2,..., k. (4.1) Since any quantum operation does not decrease the fidelity between two quantum states, for 1 i j k, we have F (ρ i, ρ j ) = F (ρ i Σ, ρ j Σ) F (ρ i ρ i, ρ j ρ j ) = F (ρ i, ρ j ) 2, so, F (ρ i, ρ j ) = 0 or F (ρ i, ρ j ) = 1. Since ρ i ρ j, F (ρ i, ρ j ) 1. Therefore, F (ρ i, ρ j ) = 0, this shows that ρ i ρ j = 0, for 1 i j k.

31 Applications Sufficiency. Suppose that for 1 i j k, ρ i ρ j = 0, then there exists a unitary matrix U such that ρ i = Uρ iu is diagonal for every 1 i k. Take Σ D N such that Σ = UΣU is diagonal (e.g., Σ = N 1 I N ) and write ρ i, Σ as a i c ρ 0 a i i =, Σ 0 c =, a i N c N respectively.

32 Applications Since (ρ i Σ )(ρ j Σ ) = (Uρ i ρ j U ) (Σ Σ ) = 0 and (ρ i ρ i)(ρ j ρ j) = (Uρ i ρ j U ) (Uρ i ρ j U ) = 0, for all 1 i j k, we know that both {ρ i Σ } k i=1 and {ρ i ρ i }k i=1 are commuting families of Hermitian matrices.

33 Applications Define M 11 M 12 M M 1N M 21 M 22 M M 2N M = M 31 M 32 M M 3N, M N1 M N2 M N3... M NN where each M ij is an N N square matrix for 1 i, j N such that M ij = 0(i j) and a i 1 a i 2... a i N a i 1 a i 2... a i N M jj = if a i j 0 for some i(1 i k); a i 1 a i 2... a i N otherwise,

34 Applications Clearly, M is a row stochastic matrix. Since ρ i ρ j = Uρ iρ j U = 0 for 1 i j k, N r=1 ai ra j r = 0 and a i r, a j r 0 implies a i ra j r = 0 for each 1 r N and 1 i j k. It follows from tr(ρ i ) = tr(σ ) = 1 that there must at least exist 1 i 1, i 2,..., i k N such that a 1 i 1, a 2 i 2,..., a k i k 0, but a j i s = 0 for all j s(s, j = 1, 2,..., k). For fixed j, there exists at most some 1 i k such that a i j = 0.

35 Applications Thus, (a 1 1 )2... a 1 1 a1 N... a1 N a (a 1 N )2 (a 2 1 )2... a 2 1 a2 N... a2 N a (a 2 N )2... (a k 1 )2... a k 1 ak N... ak N ak 1... (a k N )2 a 1 1 c 1... a 1 1 c N... a 1 N c 1... a 1 N c N a 2 1 = c 1... a 2 1 c N... a 2 N c 1... a 2 N c N M (4.2)... a k 1 c 1... a k 1 c N... a k N c 1... a k N c N is valid.

36 Applications M N 2 It follows that there exists a quantum operation φ : M N 2 satisfying φ (ρ i Σ ) = ρ i ρ i(i = 1, 2,..., k). (4.3) Let φ (ρ) = l i=1 V iρv i, ρ M N 2, where l i=1 V i V i = I. Now, we define φ(ρ) = l [(U U) V i (U U)]ρ[(U U) V i (U U)], ρ M N 2. i=1

37 Applications Clearly, l i=1 [(U U) V i (U U)] [(U U) V i (U U)] = I, so φ : M N 2 M N 2 is a quantum operation and for i = 1, 2,..., k, we have φ(ρ i Σ) = (U U) φ (ρ i Σ )(U U) = (U U) (ρ i ρ i)(u U) = (U ρ iu) (U ρ iu) = ρ i ρ i.

38 Thank you!